# Thread: Not completely understanding what analytic means. HELP!

1. ## Not completely understanding what analytic means. HELP!

I understand the following:
Theorem: Suppose the real functions $u(x,y)$ and $v(x,y)$ are continuous and have continuous 1st order partials in a domain D. If $u$ and $v$ satisfy the CR equations at all points in D, then $f(z) = u(x,y) +iv(x,y)$ is analytic in D.

Therefore, the CR equations $u_{x} = v_{y}$ and $u_{y}=-v_{x}$ must hold.

But I don't understand what I'm supposed to conclude when I do the following problem:
$f(z)=|z-10|^2 = y^2 + (x^2 - 10)^2$

When I do the CR equations I get $v_{y} = 0$ and $v_x=0$. Does that mean it is only differentiable at $(0,0)$? What does that say about being analytic? I've seen a few problems like this were some of the CR equations are equal to zero such as $f(z) = |z|^2$. For that one I understand that limits at different paths are not equal therefore only differentiable at $z = 0$. Do I need to do different paths to show where it's only differentiable for the problem above or how should I try to tackle this problem? Thank you so much!

2. ## Re: Not completely understanding what analytic means. HELP!

"Analytic" is defined at individual points- the simplest definition is that "f(z) is analytic, at $z= z_0$, if and only if it is equal to its Taylor's polynomial in some open neighborhood of $z_0$, so showing it satisfies the Cauchy-Riemann equation at z= 0 doesn't really prove it is analytic at any point. I don't know what you mean by "different paths". What do "paths" have to do with differentiability?

3. ## Re: Not completely understanding what analytic means. HELP!

Originally Posted by HallsofIvy
"Analytic" is defined at individual points- the simplest definition is that "f(z) is analytic, at $z= z_0$, if and only if it is equal to its Taylor's polynomial in some open neighborhood of $z_0$, so showing it satisfies the Cauchy-Riemann equation at z= 0 doesn't really prove it is analytic at any point. I don't know what you mean by "different paths". What do "paths" have to do with differentiability?
Okay, I've read my notes a few more times and I think I'm beginning to understand. For the problem I stated I got $u_x = 2x - 20$, $u_y = 2y$, $v_x = 0$, and finally $v_y = 0$. Therefore, $2x -20 = 0$ and $2y = 0$. Therefore, it is only differentiable at $(0,0)$. So how do I jump to the conclusion that it is (or it isn't) analytic?

4. ## Re: Not completely understanding what analytic means. HELP!

Originally Posted by HallsofIvy
"Analytic" is defined at individual points- the simplest definition is that "f(z) is analytic, at $z= z_0$, if and only if it is equal to its Taylor's polynomial in some open neighborhood of $z_0$, so showing it satisfies the Cauchy-Riemann equation at z= 0 doesn't really prove it is analytic at any point. I don't know what you mean by "different paths". What do "paths" have to do with differentiability?
And by paths I meant taking the limit along the real axis or the imaginary axis or any other point. If both limits aren't the same then the limit of f(z) does not exist.

5. ## Re: Not completely understanding what analytic means. HELP!

I've read my notes over and over again and I'm still stuck. Can anyone help me out?